The diameter and radius are easy,

直徑、半徑都好解決

they're just straight lines

只是直線

you can measure with a ruler.

用尺就行了

But to get the circumference,

但量圓周時

you'd need measuring tape or a piece of string,

就要動用到捲尺或線

unless there was a better way.

除非有更好的辦法

Now, it's obvious

很明顯地

that a circle's circumference would get smaller or larger

圓周長短會變

along with its diameter,

直徑也是

but the relationship goes further than that.

但其實背後有更深的含意

In fact, the ratio between the two,

事實上

the circumference divided by the diameter,

圓周長除以直徑

will always be the same number,

數值永遠一樣

no matter how big or small the circle gets.

大圓小圓都一樣

Historians aren't sure when or how

史學家不確定這個數字

this number was first discovered,

是何時出現、如何求得的

but it's been known in some form

但它以某種形式

for almost 4,000 years.

存在了 4000 年之久

Estimates of it appear in the works of ancient Greek,

各地數學家都有紀錄，包含希臘

Babylonian,

巴比倫

Chinese,

中國

and Indian mathematicians.

和印度

And it's even believed to have been used

大家也相信

in building the Egyptian pyramids.

埃及金字塔和這數字有關

Mathematicians estimated it

數學家用圓內接多邊形

by inscribing polygons in circles.

來估算這個數字

And by the year 1400,

西元 1400 年之前

it had been calculated to as far as ten decimal places.

已求出小數點後第 10 位

So, when did they finally figure out the exact value

那究竟何時才有精確的數字

instead of just estimating?

而不只是一個估值呢？

Actually, never!

還沒求到！

You see, the ratio

其實

of a circle's circumference to its diameter

圓周長與直徑的比

is what's known as an irrational number,

是無理數

one that can never be expressed

小數點後，位數無限

as a ratio of two whole numbers.

而且不會循環

You can come close,

雖然估值很接近

but no matter how precise the fraction is,

但不管換算的分數多精確

it will always be just a tiny bit off.

還是差了那麼一點

So, to write it out in its decimal form,

如果用小數表示

you'd have an on-going series of digits

需要一連串的數字

starting with

也就是

3.14159

3.14159

and continuing

後面還有

forever!

沒完沒了！

That's why, instead of trying to write out

所以通常不會真的這樣寫

an infinite number of digits every time,

因為永遠寫不完

we just refer to it using the Greek letter pi.

就直接用希臘字 π 來表示

Nowadays, we test the speed of computers

若想測量電腦的運算速度

by having them calculate pi,

就讓電腦計算 π

and quantum computers have been able

量子電腦很厲害

to calculate it up to two quadrillion digits.

能算出 2000 兆個數字

People even compete to see

大家也會辦比賽

how many digits they can memorize

看誰最會背 π

and have set records for remembering

目前的世界紀錄

over 67,000 of them.

多達 67000 多位數

But for most scientific uses,

但一般科學應用

you only need the first forty or so.

小數點後約 40 位就夠了

And what are these scientific uses?

π 要怎麼應用呢？

Well, just about any calculations involving circles,

基本上，圓的計算都會用上

from the volume of a can of soda

小至汽水罐的容量

to the orbits of satellites.

大到衛星軌道

And it's not just circles, either.

但不僅限於圓的計算

Because it's also useful in studying curves,

曲線計算也很需要

pi helps us understand periodic or oscillating systems

π 能計算週期和振盪

like clocks,

像鐘擺

electromagnetic waves,

電磁波

and even music.

甚至是音樂

In statistics, pi is used in the equation

統計上，π 可代入方程式

to calculate the area under a normal distribution curve,

來計算常態分佈曲線

which comes in handy for figuring out distributions

常用於數值分布的運算

of standardized test scores,

像是標準化測驗

financial models,

財務模型

or margins of error in scientific results.

或科學結果的誤差範圍

As if that weren't enough,

還不只如此

pi is used in particle physics experiments,

粒子物理實驗也會見到

such as those using the Large Hadron Collider,

像是瑞士的大強子對撞機

not only due to its round shape,

不只因為對撞機是圓形

but more subtly,

還因為一個小細節

because of the orbits in which tiny particles move.

就是微小粒子環繞的軌道

Scientists have even used pi

科學家也利用 π

to prove the illusive notion

替「光」驗明正身

that light functions as both a particle

光可以是粒子

and an electromagnetic wave,

同時也是電磁波

and, perhaps most impressively,

更厲害的是

to calculate the density of our entire universe,

π 能計算宇宙的密度

which, by the way,

不過

still has infinitely less stuff in it

整個宇宙的密度

than the total number of digits in pi.

還是比 π 無限的數字還少